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December 12, 2006

Navigating Geometric Langlands by Analogies

Posted by Urs Schreiber

On occasion of the new paper by Gukov-Witten on further details of how the geometric Langlands duality can be understood as a special case of S-duality in 4-dimensional Yang-Mills theory or, equivalently, of Mirror-symmetry of the 2-dimensional field theory obtained from that by compactification, I would like to recall some – analogies.

No, I am not an expert on geometric Langlands, and I haven’t even learned (mostly from Tony Pantev I, II and Anton Kapustin I, II, III) more than the most basic statements involved. Usually, talking about Langlands duality, classical or geometric, quickly leads to ever and ever more sophisticated constructions. It’s heavy machinery taken to its extreme.

But usually, if something is exceptionally interesting it is so not because it is arbitrarily convoluted and that being it – but because governing all that complexity is a single elegant mechanism.

Exactly that seems to be true for geometric Langlands, including its connection to quantum field theory. And it ought to be emphasized a little more.

We have a seminar here in Hamburg this semester, where the mathematicians join forces with the string theorists in the hope of teaching each other a tiny little bit of what geometric Langlands is about. In order to prepare myself for that, I want to start rather leisurely by simply recalling some analogies and simple pictures that should help discern the underlying basic structure.

The basic picture is this:

The geometric Langlands duality is about a categorified Fourier transformation.

This is mentioned by Frenkel in section 4.4 of his lecture. Also see David Ben-Zvi’s comment here. The experts certainly are very well aware of this.

Still, for mere mortals such as myself, I would like to supply this statement with a dictionary that maps various concepts to each other.

Since the Fourier transformation is a linear map between vector spaces (of functionals),
in order to understand what a categorified Fourier transformation might be, we need to have a vague idea of what the categorification of a linear map between linear spaces would be.

For the present purpose, the right notion is this: like a complex vector space is a module for the monoid ℂ\mathbb{C}, a complex 2-vector space is a module category for the monoidal category Vectℂ\mathrm{Vect}_\mathbb{C}.

The simplest example of such a module category is the category of vector spaces itself. The next simplest one is that of complex vector bundles on some space XX. Since we want to be sophisticated, we should think of these as locally free sheaves (of sections of our vector bundles). And if you have made it that far, you should allow yourself to also consider complexes of such sheaves as certain 2-vector spaces. In particular complexes as live in derived categories of coherent sheaves on some moduli stack. There might be a technical issue or two with making usefully precise how these are “2-vector spaces”, but all I am saying here is that derived coherent sheaves are nothing but a souped-up version of vector bundles, and these we easily think of as 2-vectors.

With that picture of sheaves as 2-vectors understood, the main statement of the geometric Langlands duality is easily seen to be a candidate for a categorification of the Fourier isomorphism.

Geometric Langlands Conjecture - very vague but helpful version: There exist certain derived categories of sheaves (think: 2-vector spaces) AA and BB, where BB is “spanned” in some sense by eigensheaves (think: eigen-2-vectors) of a certain Vectℂ\mathrm{Vect}_\mathbb{C}-linear endofunctor H:B→BH : B \to B (think: differentiation operator), called the Hecke operator, and where AA is similarly “spanned” by skyscraper sheaves (think: 2-δ\delta-distribution) such that there is an equivalence of derived categories
c:A→∼B
c : A \stackrel{\sim}{\to} B
(think: Fourier transformation) which sends skycraper sheaves to Hecke eigensheaves (think: which sends δ\delta-distributions to plane waves).

The rest is details. :-)

What Witten and collaborators (Kapustin and Gukov, so far) are adding to this picture is another level of analogy. As I mention from time to time # there is a nice way to think of 2-dimensional quantum field theory as a categorification of ordinary quantum mechanics. And as quantum mechanics is about linear maps between vector (okay: Hilbert-)spaces, 2-dimensional quantum field theory is about 2-vector spaces.

It’s getting late and there would be lots of things to say here (some of which I have already said before, for instance in Fourier-Mukai, T-Duality and other linear 2-Maps). Maybe I leave it at that for the moment and just finish with presenting what was the main purpose of this entry here: the following table deserves to be written down.

12 Comments & 3 Trackbacks

Re: Navigating in Geometric Langlands by Analogies

a generalization of the Fourier transform from the real line (a simple frequency axis) to the entire complex plane,

is there room for an expansion of your chart? (I see you moderated a discussion on Laplace vs. Fourier.) Then, if you say yes, I’d like to ask you for more about its relationship to the Legendre transform, as I did John.

Re: Navigating in Geometric Langlands by Analogies

Now there’s a categorifier, Mikhael Kapranov, working on a noncommutative Fourier transform in his ‘Noncommutative Geometry and Path Integrals’. A follow-up paper is advertised ‘Membranes and higher groupoids’.

Read the post Ubiquitous DualityWeblog: The n-Category CaféExcerpt: I'm in one of those phases where everywhere I look I see the same thing. It's Fourier duality and its cousins, a family which crops up here with amazing regularity. Back in August, John wrote: So, amazingly enough, Fourier duality...Tracked: January 11, 2007 2:17 PM